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Polynomial $6j$–symbols and states sums

Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 11 (2011) 1821–1860

For a given 2r–th root of unity ξ, we give explicit formulas of a family of 3–variable Laurent polynomials Ji,j,k with coefficients in [ξ] that encode the 6j–symbols associated with nilpotent representations of Uξ(sl(2)). For a given abelian group G, we use them to produce a state sum invariant τr(M,L,h1,h2) of a quadruplet (compact 3–manifold M, link L inside M, homology class h1 H1(M, ), homology class h2 H2(M,G)) with values in a ring R related to G. The formulas are established by a “skein” calculus as an application of the theory of modified dimensions introduced by the authors and Turaev in [Compos. Math. 145 (2009) 196–212]. For an oriented 3–manifold M, the invariants are related to τ(M,L,ϕ H1(M, )) defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of Uξ(sl(2)). They refine them as τ(M,L,ϕ) = h1τr(M,L,h1,ϕ̃) where ϕ̃ correspond to ϕ with the isomorphism H2(M, ) H1(M, ).

$6j$–symbols, state sum, skein calculus, quantum groups, $3$–manifolds
Mathematical Subject Classification 2000
Primary: 57M27, 81Q99
Received: 13 July 2010
Accepted: 8 October 2010
Published: 12 June 2011
Nathan Geer
Mathematics & Statistics
Utah State University
3900 Old Main Hill
Logan, UT 84322
Bertrand Patureau-Mirand
Universite de Bretagne-Sud
BP 573
F-56017 Vannes